Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #127⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C2×#384⋊C3 |
| unambiguous semidirect product | (C2×#384)⋊C3 |
| Fitting, Sylow | C2×#384 ⋊ C3 |
| derived | C42 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #127⋊C3 |
| commutator, hypocenter | C42 | C2×C3×C4 |
| Frattini | C23 | C22×A4 |
| Fitting | C2×#384 | C3 |
| socle | C24 | C2×A4 |
| prime radical | C22×C42⋊C3 | C2 |
| cosocle | C2×C42 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C22 | |
188 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4)(5,7,11)(6,9,13)(8,14,15,10,12,16);(5,6)(7,14,9,12)(8,10)(11,15,13,16)40 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#384⋊C3, #13830⋊C3, #14665⋊C3, … |
| maximal normal subgroup of | C4×#384⋊C3, #13830⋊C3, #14665⋊C3, … |
| maximal quotient of | C4×#384⋊C3, C2×#1444⋊C3, #13856⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5746);